Historia Mathematica26 (1999), 52–67Article ID hmat.1998.2218, available online at http://www.idealibrary.com on

On the Transmission of RiemannÕs Ideas to Portugal

Jeremy Gray

Faculty of Mathematics and Computing, Open University, Milton Keynes, MK7 6AA, United Kingdom

and

Eduardo L. Ortiz

Department of History, Harvard University, Cambridge, Massachusetts 02138

Henrique Manual de Figueiredo is a secondary figure in the history of science in Portugal. His name isnot recorded in any of the main biographical encyclopedias on his country, in international compilationssuch as the detailed work of May (Bibliography and Research Manual of the History of Mathematics,Toronto: Univ. of Toronto Press, 1973), or in studies on the history of science and culture in hiscountry. However, he deserves to be remembered as a unique pioneer in the transmission to Portugalof Riemann’s work, in particular of Riemann surfaces and the theory of algebraic curves. Althoughtrained within the French tradition, and on friendly terms with French scientists till the end of his life,Figueiredo as a young man turned in the direction of the mathematical ideas then being developed inGermany. His life and work are also interesting from the point of view of the study of the transmissionof science to and within peripheral countries and of their choice of foreign models. They suggestthat, far from being a slow process of regular diffusion, the transmission of mathematical ideas fromleading to peripheral mathematical communities is a complex process with selective sharp advances.Figueiredo was a respected mathematician within the structures of his own country, a professor at theUniversity of Coimbra, who held several official positions in his country and represented it at one ofthe first international encounters involving science and technology in which peripheral countries tookan active participation: the Paris Universal Exhibition of 1900.C© 1999 Academic Press

Henrique Manuel de Figueiredo ´e uma figura de segundo plano na hist´oria da ciencia em Portugal.Nao se encontra qualquer referˆencia ao seu nome, quer nas principais enciclop´edias biogr´aficas do seupaıs, quer em publica¸coes internacionais, tais como o minucioso trabalho de May (Bibliography andResearch Manual of the History of Mathematics, Toronto: Univ. of Toronto Press, 1973), nem mesmoem estudos sobre a hist´oria da ciencia e cultura do seu pa´ıs. Contudo, ele merece ser recordado comopioneiro na divulga¸cao, em Portugal, do trabalho de Riemann, em particular superficies de Riemanne teoria das curvas alg´ebraicas. Apesar da sua forma¸cao na escola francesca e de ter mantido la¸cosde amizade com cientistas franceses, durante toda a sua vida, Figueiredo enquanto jovem deixou-seinfluenciar pelas ideias matem´aticas ent˜ao desenvolvidas na Alemanha. Na sua vida e obra tiveramtambem um papel importante a divulga¸cao de ciencia em pa´ıses perifericos e o contributo para a es-colha de modelos estrangeiros. Parece que a transmiss˜ao das ideias matem´aticas dos centros principaispara as comunidades matematicas perif´ericas, longe de ser um processo lento e regular, foi antes umprocesso complexo com progressos altamente irregulares. Figueiredo foi um matem´atico conceituadonas estruturas do seu pr´oprio pa´ıs, era Professor na Universidade de Coimbra, ocupou v´arios cargosoficiais no seu pa´ıs e representou-o num dos primeiros encontros internacionais de ciˆencia e tecnolo-gia, a Exposi¸cao Universal de Paris, em 1900, na qual pa´ıses perifericos tiveram uma participa¸caoactiva. C© 1999 Academic Press

MSC 1991 subject classifications: 01A55; 01A60; 01A70.Key Words: Riemann; Portuguese mathematics; Henrique Manuel de Figueiredo; Riemann surfaces.

520315-0860/99 $30.00Copyright C© 1999 by Academic PressAll rights of reproduction in any form reserved.

HMAT 26 RIEMANN’S IDEAS IN PORTUGAL 53

1. HENRIQUE MANUEL DE FIGUEIREDO

Figueiredo was born in Coimbra on 13 August, 1861 and died in the same city on 24February, 1922. He came from a family of distinguished Portuguese intellectuals. His fa-ther, Manuel Adelino Figueiredo, was a graduate of the Faculty of Philosophy, University ofCoimbra, and became interested in the problems of Portuguese agriculture. The year of hisson Henrique’s birth he published hisEstudos de agricultura, a study on that topic whichhad some impact in his country and abroad. From 1861 till 1865 he was commissioned bythe government to work in Porto. He contracted tuberculosis and died young in January1865, when his son was three years old. The education of Henrique was the responsibilityof his mother, J´ulia Aillaud Monteiro de Figueiredo, and of his grandparents, who had aconsiderable influence on him.

In 1836, two years after the end of the Portuguese civil war, a much needed reformwas made in Coimbra. The teaching of mathematics had remained almost unchanged sincethe important reforms of Marques de Pombal, in 1772, when teaching of the sciences wasincorporated into the university. A group of young enthusiastic teachers joined Coimbra’sFaculty of Mathematics as professors of mathematics. One of them was Abilio Affonsoda Silva Monteiro, the father of J´ulia. He belonged to a generation of mathematicianswho combined teaching with other activities, related mainly to government and to theadministration of technical state offices.1

Their contributions to mathematics centered on teaching and the production of more mod-ern textbooks. Most of their “research” centered on observations suggested by their study ofthe books they used in their teaching. These were often translations of established foreigntextbooks, mainly French. In some cases the translations were free or based on more thanone source, omitting and adding material from its references to make them better adapted tothe needs of local teaching. In this last case the sources used might not necessarily be men-tioned explicitly. This procedure was widely used in a number of other peripheral countries,such as Spain and Argentina, and may be true for countries outside the Iberian world.

On her mother’s side, J´ulia belonged to the family of the owners of the publishing houseAillaud et Cie, founded in Paris by her grandfather Jean Pierre Aillaud. The paternal grand-father of Henrique was another distinguished person in contemporary Portugal: ManuelMarques de Figueiredo was also a professor at the University of Coimbra, but he was in thePhilosophy Faculty. He had been a president of the Senate of Coimbra and had taken partin works for the welfare of the underprivileged of his home town. The two grandfathers andthe father of Henrique were Knight Commanders in different Portuguese orders; the latterhad religious as well as social connotations.

These two influences, mathematical and philosophical, can be easily detected inHenrique’s career. After very successful studies at school he enrolled in the Faculty ofMathematics at Coimbra University in 1879, at 18 years of age.

The modernization of teaching at Coimbra and the general spirit of renovation in Por-tuguese society in these years started to produce some interesting results in several fields.The mathematical horizon of Portugal began to be dominated by the most important of its

1 Abilio Affonso da Silva Monteiro was born in Ventosa do Bairro (Mealhada) 19 September, 1812 and diedin Coimbra 15 June, 1890. Besides personal files in national archives, information on Coimbra’s teachers can befound in the comprehensive work [24].

54 GRAY AND ORTIZ HMAT 26

late 19th-century mathematicians: Francisco Gomes Teixeria,2 a graduate from Coimbrawho was 10 years older than Figueiredo. He published in many of the leading mathematicsresearch journals of his time in Europe and in America and was the author of innovativetextbooks which accelerated intellectual change in his country, particularly in the field ofmathematical analysis. This was one of the areas of mathematics regarded as being of con-siderable interest because of its engineering applications. Analysis took longer to developfirm roots in Spain than in Portugal, and even there the influence of Gomes Teixeria isundeniable (see [14]).

Only two years before Figueiredo joined Coimbra, Gomes Teixeira had started publi-cation of the first Portuguese mathematical journal,Jornal des sciences mathematicas eastronomicas. This journal was initially conceived as an intermediate journal, that is, asa journal for teachers training students in the final grades of secondary school and earlyyears of university (see [17]). As such it was a contribution, from the field of mathematicsteaching, to the then national task of promoting professionalism in Portugal. This was amajor problem confronted by Portuguese society and by the state, which was making effortsto incorporate Portugal into international business and commercial networks. Perhaps forthat reason, theJornal received government support. However, Gomes Teixeira graduallychanged the aims of his journal, elevating its contents and focusing it in the direction ofmathematical research, while keeping extensive sections in which he reviewed new foreignand national books and also research papers published in other journals. Gomes Teixeira’sJornal became the research journal of Portuguese mathematicians and had a profoundinfluence on the development of mathematical journalism in the entire Iberian world.

2. FIGUEIREDO’S STUDIES IN COIMBRA

Figueiredo graduated from Coimbra as aBacharelin June 1883 with high marks.3 He waspart of a group of students who would soon become leading members of the scientific staffof that university and contribute to its further modernization. He received the second prizein the first, second, third, and fifth years of his studies and also special prizes in physics,mineralogy, and botany. Figueiredo had also registered as a student in the PhilosophyFaculty, and graduated there as aBacharelon 8 July 1882.

In the summer of 1884 he decided to do further studies in Paris, where part of hismother’s family resided. French language and culture were still dominant in Portugal. Thiswas, however, a time when some Portuguese intellectuals began to develop a more criticalattitude towards French culture. The leading Portuguese writer, E¸ca de Queiros, wouldlater publish a brilliant essay [20, 383–411] on this matter. In it he described, with unusualpenetration, the extraordinary extent of French influence in Portugal, from primary schoolto the highest layers of culture and also in ordinary life.

Figueiredo intended then to make a career in engineering and passed the entrance exam-inations for theEcole nationale de mines, in Paris. Soon after the beginning of his courseshe became seriously ill and was advised to leave Paris, as the climate there did not suit hiscondition, and return to Portugal. Most probably his family feared a repeat of his father’s

2 Francisco Gomes Teixeira (1851–1933) specialized in series expansions, interpolation, and the theory ofcurves.

3 He collected 85% of full marks.

HMAT 26 RIEMANN’S IDEAS IN PORTUGAL 55

fate. A. Daubree, a member of the Institute of France and honorary director of theEcole,may have given him support in this critical period of his life.

On his return to Portugal, as his health improved, he decided to continue his studies inCoimbra. At the end of 1884, he was invited to join the Institute of Coimbra, a learnedsociety which did much to establish a bridge with contemporary European culture; he waselected member on 13 December, 1884. It published an important literary and scientificjournal calledO Instituto, which occasionally included papers on mathematics.4 On 14April, 1886, he graduated asLicentiatein mathematics and on 10 July the same year healso graduated as aBacharel Formado5 in Philosophy. He immediately started working forhis doctorate, which he received on 6 November, 1887. The topic he chose was Riemannsurfaces.

3. FIGUEIREDO’S DOCTORAL THESIS: RIEMANN SURFACES

Graduation in Coimbra was a solemn public occasion, presided over by the Dean of theFaculty of Mathematics, who was then the mathematician Luiz da Costa e Almeida.6 Thecandidate had the privilege of choosing a patron, who introduced him; usually this wasone of his mentors. Figueiredo asked his grandfather, the veteran mathematician, AbilioAffonso da Silva Monteiro, to perform that duty for him. Monteiro had retired from theFaculty of Mathematics long before, in 1869.7 The Dean read an oration [12] in which heexalted the merits of the candidate.

A doctoral thesis examination in Coimbra consisted of two parts: the first was a selec-tion of topics in several areas of mathematics on which the candidate made (or answered)statements which he had to defend; this was a sort of general testing of the candidate’scommand of the main topics he had studied. Then there was the usual dissertation on a fixedtopic. For the first part, Figueiredo defended statements in several fields. On algebra, therewere two statements on the theory of equations, one on numerical equations, and the otheron a procedure more efficient than elimination; on calculus again two statements, one aquestion on the convergence of series and the other on abelian integrals; there were also twostatements on descriptive geometry, on the principle of continuity, and on duality. Therewere four statements each for mechanics, astronomy, and mathematical physics. The firsttwo on mechanics related to fluid mechanics and the other two to theoretical mechanics.One of them considered cases in which imaginary values for a time parameter may have amechanical interpretation, which indicates an interest in widening the acceptability of com-plex numbers. In astronomy, three statements were on instrumentation and one on comets;on mathematical physics, the statements concerned heat, potential theory, thermodynam-ics, and the behavior of Poisson’s elastic plates. In geodesy and celestial mechanics, therewere three statements each. The statements on geodesy were on triangulation, topographicprojections, and the determination of the shape of the earth; for celestial mechanics, one

4 In its first 90 volumes from 1852, this journal published no less than 70 papers on mathematics.5 This degree was roughly equivalent to that ofLicentiate.6 Luiz da Costa e Almeida produced a number of booklets on determinants, calculus of variations, and geometric

aspects of complex numbers used for the renovation of teaching in the last quarter of the nineteenth century inPortugal. He also wrote on mechanics (with applications to projectiles) and on the institution of which he was Dean.

7 Monteiro received his doctorate in 1838. At the University of Coimbra he taught astronomy, celestial mechan-ics, advanced algebra, analytic geometry, advanced calculus, descriptive geometry, and mathematical physics.

56 GRAY AND ORTIZ HMAT 26

was on methodology, the second on lunar theory, and the last on cosmogony. This part of hiswork was dedicated to his grandparents. It gives a fairly clear picture of the training givenin Coimbra in the 1880s. The preponderance of topics related to astronomy, geodesy, andtopography is clearly related to the needs of Portugal in the fields of engineering, navigation,and land surveying.

The topic chosen by Figueiredo for his dissertation was, as we said, an exposition onRiemann surfaces. His is a monograph of some one hundred pages which he dedicatedto his mother. He stated that his monograph was based on two of Riemann’s works: theGrundlagen fur eine allgemeine Theorie der Funktionen einer veranderlichen complexenGroβeof 1851 and theTheorie der Abel’schen Funktionenof 1857.8 He also indicated thathe adopted the point of view of Puiseux [18] in his work.

The choice of this topic which, as Figueiredo remarked, was then mainly pursued in Ger-many, is an interesting departure from the mathematics of French authors and a movementin the direction of that of Germans. A similar departure may be noted in Spain in connectionwith the numerical solution of algebraic equations in 1879 (see [15]). These choices cannotbe separated from the emerging critical attitude of French thought in philosophical and liter-ary circles in Portugal and other peripheral countries at the time and from the re-evaluationswhich followed the defeat of France by Prussia in 1870.9

Figueiredo’s inaugural dissertation was published in 1887 by the University Press ofCoimbra, with the titleSuperficies de Riemann[6]. It is a small work of 103 pages thatis now difficult to find, and it is noteworthy that an exposition of Riemann’s ideas shouldhave been given in Portugal at the time, some years before the first systematic treatises inEnglish, for example.

The Superficies de Riemannis divided into two parts, as Figueiredo mentioned in hisintroduction. An introductory 28 pages study how the leaves of a Riemann surface arepermuted in the neighborhood of a critical point. Then a long section of 38 pages is givenon the topology of a Riemann surface; its definition, the concept of a boundary on a Riemannsurface, and the idea of multiple connectedness. There is a much shorter section on the ideaof an integral on a Riemann surface, and the dissertation closes with the interesting specialcase of elliptic integrals seen from this perspective.

Chapter 1 considers an algebraic function defined by an irreducible polynomial equationF(y, x)= 0 in two complex variablesx and y, of degreen in y andm in x. Figueiredoattributed the usual geometrical representation of a complex number as a point of the planeto Cauchy, and took one plane for the variablex and another for the variabley. He thenraised the question of the behavior of the values ofy in the neighborhood of a critical point,defined as a point where∂F

∂y = 0 (Figueiredo wroted Fdy ). A separate examination, Figueiredo

pointed out, must be made for the points where at least one of the variables is infinite. Heobserved that the values ofy depend continuously onx but are the same along two pathsof x provided the paths can be deformed into one another without crossing a critical pointor becoming infinite.

8 These two works have been reprinted in [23, 3–48, 88–114]. Figueiredo may have used the 1876 edition ofRiemann’s collected works [22].

9 The 1870 defeat produced a mass of literature by local scientists critical of the way in which science, technology,and their relation to the war effort had evolved in France. Such development was often contrasted with that of therevolutionary period. See [16].

HMAT 26 RIEMANN’S IDEAS IN PORTUGAL 57

To understand the variation between paths that together enclose one or more criticalpoints. Figueiredo first considered the case where, whenx= α, there arep common valuesof y, sayy=β. Whenx is close to but distinct from the valueα, these values ofy will allbe distinct. Now, as Figueiredo showed, asx traces an infinitely small circle aroundα, thep values ofy, sayβ + y1, . . . , β + yp are permuted.10 To examine this case in more detail,Figueiredo first considered the case where∂F

∂x 6= 0, so the point in question is not a singularpoint, and showed by suitable changes of variable that thep values ofy are permutedcyclically asx describes a circuit aroundα. The case where∂F

∂x = 0 is more complicated,and, in [6, 13], Figueiredo introduced the polygon method earlier described by Puiseux andtoday often called the Newton polygon. He showed that this permits one to determine whichsets of values ofy are permuted cyclically among each other. Finally, the case where eitherx or y or both is infinite is discussed, using the change of variablesx′ = 1

x , y′ = 1y .

In Chapter 2, Figueiredo introduced the concept of a Riemann surface by means of anexample: the algebraic functiony of x defined by the equation

F(y, x) = y2− A(x − a1)(x − a2)(x − a3)(x − a4) = 0.

It yields a two-valued function wheneverx 6=a1,a2,a3, or a4; the two function valuesare equal but opposite. At the pointsx=a1,a2,a3,a4, y= 0. At no point does∂F

∂x = 0,so asx goes through a small circle arounda1, say, the corresponding values ofy arepermuted cyclically. Figueiredo called the pointsa1,a2,a3,a4 ramification points,pontosde ramificac¸ao [6, 31]; later [6, 34], he gave the German termWindungspunkt. He thendrew in thex-plane a straight line from each pointx= a1, . . . ,a4 to an arbitrary pointx= q and explained how they values were distributed in two leaves (Folhas[6, 33]) whichwere permuted at the ramification points as the moving point in thex-plane crossed oneof the linesaq. The resulting surface is an example of a Riemann surface (superficie deRiemann) which, because of the behavior of the leaves was, he added, sometimes called aWindungsflachein German. Figueiredo also admitted other examples of Riemann surfaces,such as parts of the surface associated to an algebraic function, discs, annuli, and regionscontaining one or more branch points.

His opening example led him to the general case, the Riemann surface correspondingto the algebraic equationF(y, x)= 0. This is the geometrical locus of all points, criticalpoints included, that satisfy the equation. The local behavior had been analyzed in Chapter 1;now [6, 36], Figueiredo introduced the term ramification point of orderθ to describe thesituation whereθ + 1 leaves are exchanged cyclically at a point on the Riemann surface.For simplicity, and without loss of generality, Figueiredo now assumed that the surface hadno ramification points at infinity.

To analyze the complicated nature of a Riemann surface, Figueiredo turned in the secondpart of Chapter 2 to topological considerations. He called a surface, or a part of a surface,connected (connexa) when any two points in it could be joined by a path lying entirely inthe surface. In practice, he said, all curves would be drawn to miss the singular points. Aclosed surface (superficies fechadas[6, 39]) was, by definition, one that divided space into

10 Figueiredo wrote “uma circumferencia de raio infinitamente pequeno,” in keeping with the language of theday, for a circle of arbitrarily small (but finite) radius which contains no other branch points.

58 GRAY AND ORTIZ HMAT 26

two disconnected pieces.11 He then proposed to classify surfaces according to the type andnumber of different curves that they could possess.

He said that a point lay on the boundary of the Riemann surface if it could not besurrounded by a curve that could be shrunk arbitrarily close to the point. The set of suchpoints formed what Figueiredo called the total boundary of the surface (limite total [6, 40]).A curve (or curves, in Figueiredo’s later usage) which divides the surface into two partssuch that any path joining a point in one part to a point in the other crosses the curve is calleda complete boundary (limite completo). For example, on the Riemann surface consisting ofthe double cover of the plane branched over two points, a circle surrounding both branchpoints is a complete boundary. On a surface with a total boundary, an incomplete boundary(limite nao completo[6, 42]) is a closed curve that has the following property: points that arearbitrarily close to the curve but lie on either side of it can either be joined by a continuouscurve that does not cross the given closed curve, or they can be joined in this way to thetotal boundary. For example, the Riemann surface formed by the points{z: 1< |z|< 3} hasthe curve|z| =2 as an incomplete boundary. Figueiredo then indicated that the distinctionbetween complete and incomplete is between a curve (or curves) which can form a totalboundary of a piece of the given Riemann surface and ones which cannot.12

If a system ofn curves can be drawn on a Riemann surface to form a complete boundary,it will not be unique. But topological significance attaches to the number of curves in acomplete boundary. To explain this idea, Figueiredo began (Sect. 26) by observing that anyfamily of complete boundary curves on a Riemann surface divides the surface into regionsone may shade black and white like a chess board. Two points are given the same colorif any path joining them crosses the given curves an even number of times. A system ofincomplete boundary curves he called an incomplete system of boundary curves if it waspossible to join two neighboring points on either side of one of the curves by a path thatdoes not cross any of the curves in the system. He gave the example of a Riemann surfacedefined as a double covering of the plane branched over three points,a, b, andc, say. Hetook three circlesA, B, andC such thatA containedb andc but nota, B containedc andabut notb, andC containeda andb but notc. Then any pair of circles formed an incompletesystem of boundary curves. (To be precise, this was Figueiredo’s intention, but the diagram[6, 44] accompanying his argument is flawed.)

Suppose that a system ofn incomplete boundary curves is given, anyn− 1 of which forman incomplete system of boundary curves. Suppose, moreover, that then curves separatethe surface into two path-connected regions. Then, said Figueiredo, the system ofn curveswill be called a complete irreducible boundary system. For example, the curvesA, B, andC of the previous example form such a system.

Figueiredo then compared three systems of closed curves, say (A), (B), (C), no curveof which passes through a branch point, and supposed that the systems (A, B) and (A,C)

11 Without making it clear, Figueiredo seemed to be thinking of the usual, intuitive embeddings of Riemannsurfaces into 3-dimensional space, for which indeed the ones without boundary (“closed” in modern terminology)do separate space into a region inside and another outside. Surfaces which are not closed, in the modern sense,have boundaries which span holes connecting the “inside” and the “outside.”

12 In modern terminology, a complete boundary is homologically trivial, whereas an incomplete one is not. Thedistinction between homology, which Figueiredo used, and homotopy, goes back to Riemann.

HMAT 26 RIEMANN’S IDEAS IN PORTUGAL 59

are both complete boundary systems. He then showed by a simple parity argument that thesystem (B,C) is also a complete boundary of a region on the surface.

Figueiredo, following Riemann, called a surface simply connected (simplesmente con-nexa) if any simple closed curve forms a complete boundary of part of it, and said any otherkind of surface was multiply connected (connexao multipla). It follows from what he hadalready proved that there is a maximum numbern such thatn curves can be found whichneither separately nor together form a complete boundary system, and the numbern+ 1 hecalled the order of connectivity of the surface. He commented [6, 54] that “This definition,which is based essentially on the arbitrary choice ofn closed curves, is perfectly generaland precise.” It shares with Riemann’s approach the tacit assumption that there are alwaysa finite number of curves having the required property.

One of Riemann’s profound ideas had been to use systems of curves to change a given mul-tiply connected Riemann surface into a simply connected region by dissection. Figueiredonext explained how this can be done. He defined the termQuerschnittto be a curve thatjoins boundary points [6, 56]; it lowers the order of connectivity by 1. He observed that it isalways possible to draw aQuerschnitton a surface of higher connectivity (by means of tacitassumptions about points at infinity) and that the number ofQuerschnitteis always odd. Heillustrated the theory so far with an example he was to return to later in the book: a Riemannsurface of two leaves with three simple branch points and order of connectivity equal to 3.In keeping with his earlier advice to keep matters simple by allowing no branch points atinfinity, he removed the point (∞,∞) (which if admitted would have to be a simple branchpoint), surrounding it instead with a closed curve that was the total boundary of the surface.

To study integrals on a Riemann surface, Figueiredo observed that if∫ ∫ (dV

dx− dU

dy

)dx dy

makes sense in a region, then it equals∫

(U dx + V dy) taken round the boundary. Thisis a special case of Green’s theorem, not proved here. It follows that a complex integralon a Riemann surface, taken round a complete boundary of a region where the functionremains finite and continuous, vanishes. This is the Cauchy integral theorem in the settingof Riemann surfaces. If there are points of discontinuity, then the integral reduces to theintegral taken around them. So, by a system ofQuerschnittegoing round any “poles,”an integral on a Riemann surface can be made into a single-valued function of its upperendpoint on the region bounded by the cuts.

The integral of a function defined on a closed path is called an elementary period ofthe function. Figueiredo showed that it is well defined, i.e., that it is the same on any twopaths that can be deformed into each other without crossing a branch point of the Riemannsurface or a pole of the integrand. He then gave the important but simple example of ellipticfunctions, interpreting them as integrals of an everywhere finite integrand on the curvedefined by his opening example:

F(y, x) = y2− A(x − a1)(x − a2)(x − a3)(x − a4) = 0.

He showed that such an integral taken between two points is well defined up to periods,

60 GRAY AND ORTIZ HMAT 26

and established in detail the traditional connection with elliptic integrals. He explained thedouble periodicity of the functions and the meaning of the complete elliptic integrals, thusshowing how Riemann’s geometrical approach to complex function theory made sense ofone of the best established analytic theories of the period.

Despite the fact that Portugal was closer intellectually to France than Germany at thistime, the most likely sources for this information seem to be German. The topology isinevitably very like Riemann’s paper on Abelian functions, which does compare systemsof curves, but the parity argument seems to be original to Figueiredo. The most confusingpart of the book is the discussion of boundaries. In the absence of specific references towhat Riemann wrote, it is likely that Figueiredo was attempting a modest synthesis of twoapproaches that had been taken by Riemann. In his 1851 dissertation, Riemann workedwith surfaces defined over bounded compact regions of the complex plane, and whichtherefore have boundaries. In his 1857 paper, he considered compact surfaces defined overthe compactified plane; these do not have boundaries. For example, deleting a branch pointat infinity is in keeping with the earlier style, whereas many of the definitions, and the laterchoice of examples, fit more happily to the later style.

After the Puiseux material, all of what Figueiredo wrote in [6] except the parity argumentcan be found in the first (1864) edition of Heinrich Dur`ege,Elemente der Theorie derFunctionen einer complexen veranderlichen Grosse[5]. Although Durege in a book of228 pages does much more, the comparison is very close on the overlap, which includesthe treatment of elliptic functions. This had, of course, been a motivating example forRiemann himself, and his lectures on the topic had been one of Dur`ege’s sources for hisbook. Figueiredo was therefore working in a genuinely Riemannian tradition throughout.His usual restriction to double coverings enabled him to describe much of the materialwithout entering into undue complexity, and of course left him able to connect these newideas with what was still their most useful application, the theory of elliptic functions.

The book [13] by Carl Neumann, which first appeared in 1865 and came out in a muchrevised second edition in 1884, is a much less likely source. It has a wider range of examples,emphasizes the utility of the Riemann sphere, enters the theory of theta functions, and isplainly aimed at the much more difficult theory of Abelian functions. For these reasons, it isimplausible that Figueiredo derived his treatment from Neumann’s, although he may haveread it. Durege’s treatment, on the other hand, was well received in its day—the book ranto four editions—and lends itself to being cut down for Figueiredo’s purposes.

For all the closeness of Portugal to France, French sources appear even less likely. Theforeign terminology is all German, whereas the French at this time had their own terminol-ogy. The first edition of Briot and Bouquet’sTheorie des fonctions doublement periodiqueset, en particulier, des fonctions elliptiques(1859) [2] is resolutely in Cauchy’s spirit andungeometrical. The wholly different second edition of their book, marked by a new title,Theorie des fonctions elliptiques(1875) [3], is still very much more of a book on complexfunction theory than one on Riemann surfaces. It describes the Clebsch–Gordan theory offundamental loops on a Riemann surfaces, but not the theory ofQuerschnitte, and there isno direct influence from Riemann.

Bertrand’sTraite de calcul differentiel et de calcul integral (1870) [1] is a possible in-fluence. His emphasis in describing complex function theory, which was not Figueiredo’s,was on its use in evaluating real integrals, much as Cauchy’s had been many years before.

HMAT 26 RIEMANN’S IDEAS IN PORTUGAL 61

But Bertrand did confront the question of many-valued functions, and after acknowledgingthat the ideas of Cauchy, as developed by Puiseux, were clear and satisfactory, he saidthat the approach of Riemann was to be preferred. Therefore, he gave a brief outline ofmany-valued functions, explaining the idea of a Riemann surface with reference to suchmany-valued functions asy=√(z− a)(z− b)(z− c). In the final section of the secondvolume, there is an extensive treatment of elliptic functions which drew on these ideas, so,despite its brevity, Bertrand’s account may be the first one in French to present the meritsof the idea of a Riemann surface. But for clarity and rigor, Dur`ege’s account is much to bepreferred.

4. THE THEORY OF ALGEBRAIC CURVES

In 1888 Figueiredo wrote a monograph of some 65 pages on plane algebraic curves, theCurvas planas algebricas[7]. This work was submitted to a committee in charge of theselection of a substitute teacher in mathematical physics; Figueiredo was chosen for thatposition. In this monograph, Figueiredo again made reference to Riemann’s work of 1857,when dealing with the concept of genus, and correctly pointed out that it was Clebsch whotook advantage of this concept for geometry on curves and function theory on the associatedsurface (cf. [4]). Later in his monograph, he indicated that the concept of genus can be usedto classify curves; he closed the monograph by indicating the relation of some of theseresults to problems in integral calculus.

The German influence is prominent in Figueiredo’s second book. He began by defininghomogeneous coordinatesx1, x2, x3 in the plane and relating them to the familiar Cartesianplane coordinatesξ, η: they are proportional to the distance of the point from the three sidesof a triangle. He then gave the equation of a line in homogeneous coordinates and definedline coordinates.

In Chapter 2, on plane algebraic curves, he defined such a curve as the points whichsatisfy an equation of the formu= 0, which is of ordern in homogeneous coordinates. Heobserved that it therefore has1

2(n+ 1)(n+ 2) terms and that it is generally determined by12(n+ 1)(n+ 2)− 1= 1

2n(n+ 3) points, but not always. For example, two curves of ordern pass throughn2 common points, of which12n(n+ 3)− 1 are arbitrary, but the remaining12(n− 1)(n− 2) are determined. He then stated and proved the theorem that ifnp of thosepoints lie on a curve of orderp, the rest lie on a curve of ordern− p. (We shall call this theresiduation theorem; Figueiredo gave it no special name.)

Arguably the oldest concept in projective geometry is that of pole and polar. The polarof a point with respect to a conic is the line meeting the conic where the tangents from thepoint touch the conic. For a curve of ordern there is a succession of higher order polars, ofwhich1y(ux) is the first polar. This is defined to be

1y(ux) = y1∂u

∂x1+ y2

∂u

∂x2+ y3

∂u

∂x3.

Proceeding in this way,1iy(ux) is the i th polar of y. Their analytic significance derives

from their appearance in the Taylor series expansions for the functionu. Their geometricsignificance, which follows from the analysis, is what Figueiredo proceeded to explain.

62 GRAY AND ORTIZ HMAT 26

If the values ofx1, x2, andx3 are fixed and the point (x1, x2, x3) lies on the curveu= 0,then the equation1y(ux)= 0 is that of the tangent to the curve at the specified point. Thismakes it clear that exceptional properties of the curve will occur at its singular points, thosewhere all its first partial derivatives vanish:

∂u

∂x1= ∂u

∂x2= ∂u

∂x3= 0. (1)

(Figueiredo used the concept but not the term.) At such points the curve fails to have aunique tangent direction. They are picked up, Figueiredo showed, as the points at which thediscriminantR vanishes, where the discriminant is defined to be the expression obtained byeliminating the variablesx1, x2, andx3 from the equationsu= 0 and (1). If a curve has a dou-ble point, then either the self-intersection is proper, meaning that the curve has real tangents;or it is what Figueiredo called reverse and we call a cusp (where the direction of the tangentreverses); or it is what he called a conjugate point, where the tangents are purely imaginary.

A number of classical results were then derived. For a curve of ordern, the residuationtheorem shows that the maximum number of double points is1

2(n−1)(n−2). The interestingcase where two curves of ordersm andn meet in some double points was also described byFigueiredo. The quantityp= 1

2(n− 1)(n− 2)− d− r was introduced and called the genusof the curve. It was described as the difference between the maximum number of doublepoints a curve of a given degree may possess and the number it actually has. Figueiredoadded that it had been introduced by Riemann and then applied, principally by Clebsch,to the theory of curves, but he made no mention of the fact that it is a projective, indeedtopological invariant. Ak-fold point was defined as one through which passk branches ofthe curve, and shown to be the union of1

2k(k− 1) double points.Multiple points of orderk formed the subject of Chapter 5. The Hessian was introduced,

defined as the determinant of second order partial derivatives of the functionu. It had beenused by Hesse to study inflexion points of a curve. They satisfy1y(ux)= 0=12

y(ux). Itfollows, as Figueiredo showed, that the Hessian of a curve meets the curve in its inflexionpoints. He also showed that it has the same tangents as the original curve at the doublepoints and that it has a triple point at each original cusp (and tangents double up).

Figueiredo then returned to the topic of polars and what are called harmonic centers. Thefirst polar of a pointP with respect to a curve is the curve with the equation1y(ux)= 0,where the coordinates of the pointP are (y1, y2, y3). Figueiredo showed that it passesthrough the double points of the curve, and touches the curve where the curve has cusps.

He derived the familiar connection between, on the one hand, a curve and its first polarwith respect to a point and, on the other hand, the tangents to the curve from that point. Hedefined the class of a curve, which is the number of tangents that can be drawn to it from anarbitrary point, and showed that for a general curve of ordern (that is, in the nonsingularcase) the class isn(n − 1), but for a curve havingd double points andr cusps, the classreduces tok= n(n− 1)− 2d − 3r . He then derived the two Pl¨ucker formulae for a curveand the corresponding formulae for its dual.

In the final chapter, Figueiredo studied rational plane curves, or unicursal curves, those forwhich the genusp= 0. A rational curve of ordern has the maximal number of doublepoints, and the maximal number of cusps is3

2(n − 2). He observed that such curves can

HMAT 26 RIEMANN’S IDEAS IN PORTUGAL 63

be parameterized by rational functions and that therefore there is ann− 1 correspondencebetween the points of the complex line and the curve. That is, to each complex value ofλ

there is a unique point on the curve, but for each point on the curve there may ben values ofλ. However, rational curves may be uniformized by rational functions; that is, it is possibleto find ann− 1 map of the complex line to itself which introduces a new parameterizationof the curve which is a 1–1 correspondence between the complex line and the curve. Thisforms the content of L¨uroth’s theorem which Figueiredo did not prove here, contentinghimself with a reference to L¨uroth’s paper [10]. On the final page, he gave a brief indicationof how this helped with the integration of rational functions.

All of this material, except the formula for the maximum number of cusps, can be foundwhere Figueiredo doubtless learned it, in the famous textbook by Clebsch and Lindemann,Vorlesungenuber Geometrie, of 1876 [4]. The material on homogeneous coordinates isthere, including the example, in [4, 62–71]. The generalities about the number of pointsneeded to determine the coefficients of a curve are in [4, 305 ff.]; the theorem is later[4, 425 ff.]. Poles and polars naturally loom large in Clebsch and Lindemann, as does theclassification of double points and the definition of genus, together with the brief referencesto Riemann and Clebsch. So does the material on rational curves [4, 883–903], with theexception of the estimate of the maximum number of cusps. None of this means anythingother than that Figueiredo recognized that the book by Clebsch and Lindemann was simplythe best source for this information. The only rival would have been the second edition ofSalmon’sHigher Plane Curves, available to Figueiredo either in the German translation, byWilhelm Fiedler, or in a French edition, translated by O. Chemin [25]—there is no evidencethat Figueiredo could read English.

The monograph [7], which is the last mathematical work of Figueiredo, was dedicatedto Professor Daubr´ee, the director of the ParisianEcole nationale des mines who, we maysuppose, with his mother and grandfathers, was a main influence in his life.

The theory of curves is also a topic to which Gomes Teixeira made some of his mostrelevant contributions. His book on remarkable curves [27] is still a valuable reference andavailable in paperback. It is possible that Gomes Teixeira may have had an influence in thischoice of topic. It should also be remarked that Gomes Teixeira visited Italy in 1877. Atranslation of Riemann’s inaugural dissertation into Italian was made by his Italian friend,Enrico Betti, some 20 years earlier, in 1859, in the same period in which Betti publishedresearch on Abelian integrals.

Gomes Teixeira reviewed Figueiredo’s two pieces in hisJournal[28]. Of the monographon Riemann surfaces he said that Figueiredo’s work is an exposition of the method developedby Riemann for the study of multiform functions; he indicated that although this is adifficult method, which explains why it is not more generally used, it has originated workof the “highest interest.” Therefore: “Mr. Figueiredo has chosen well taking this topic forhis inaugural dissertation and has contributed to making this beautiful doctrine known inPortugal” [28, 24].13

Gomes Teixeira’s remarks clearly place the work of Figueiredo in the context of theviews of the incipient Portuguese mathematical community of the time: it is not an original

13 A departure from strictly French mathematics, even if sometimes based on French versions, can also bedetected in Argentina, where the British-trained mathematician, Valent´ın Balbın, published in 1887 a treatise onquaternions, reviewed by Gomes Teixeira in the same volume VIII of hisJornal.

64 GRAY AND ORTIZ HMAT 26

contribution, even if one may find interesting remarks and even valuable additions to theworks on which it is based. In any case, Figueiredo and his mentors clearly distinguished be-tween the important and the trivial in the international mathematical literature, particularlyat a time when, outside Germany, Riemann’s ideas were still in the process of becomingthe dominant ones. This was not a small feat.

5. THE ACADEMIC CAREER OF FIGUEIREDO

Figueiredo remained a substitute teacher in mathematical physics for nine years, until1897, when he was made a substitute teacher in advanced algebra and then a professor in1900. He retained this position until 1902, when he was made a professor of mathematicalphysics, a position he held until 1911. In that year, the faculties of mathematics and philos-ophy merged into the Faculty of Sciences.14 There, he continued as professor of mechanicsand astronomy until his death in 1922. In 1917–1918, he was professor of mathematicalphysics and also of probability calculus [24, 256, 304]. His interest in probability may be re-lated either to astronomy, which was part of his university teaching, or perhaps, more likely,to work on insurance and lotteries. Probability theory was then also attracting the attentionof artillery officers in Portugal and Spain. Besides his academic activities, Figueiredo wasvice-president of Coimbra’s municipal council between 1890 and 1892.

6. FIGUEIREDO AND THE UNIVERSAL EXHIBITION OF PARIS OF 1900

Despite his efforts to publicize in his country the work of German mathematicians,Figueiredo kept close ties with French scientists and was a member of the Soci´ete as-tronomique de France and the Soci´ete francaise de physique. From the 1890s, and partly inpreparation for celebrations of the end of the century, Portugal (and also other peripheralcountries, such as Russia, Japan, Spain, Argentina, and Mexico) started to be invited to takepart in international scientific organizations and also in international meetings of all sorts.Projects in geodesy and astronomy were then one of the crucibles of international science.

Figueiredo was requested by the government of his country to join the group takingresponsibility for the organization of the Portuguese section for the Exposition universellede 1900 in Paris.15

One of the most mathematically relevant contributions of Portugal to that meeting wasthe compilation of a catalogue of mathematical publications in Portugal which followed thenorms laid down by the Congr`es international de bibliographie des sciences math´ematiquesof 1889 in Paris, at which Gomes Teixeria was the representative of Portugal. This com-pilation was made by his colleague, Rodolfo Guimar˜aes,16 with the help of several of hismost distinguished colleagues, among them Gomes Teixeira and the astronomer F. Oom,

14 This is now the Faculty of Sciences and Technology.15 In the “Liste de MM. les membres des commissions ´etrangers pr`es de l’Exposition universelle international

de 1900, Paris, 1900,” pp. 30–31, Figueiredo appears as an Attach´eea la Commission executive. Figueiredo playedalso a minor role in the Exhibition itself, acting as a member of the International Jury in the award of some prizes:Jury international, Exposition universelle internationale de 1900, Paris, 1900, Section XIII, groupe 80.

16 Rodolfo Guimaraes (1866–1918) studied at the University of Porto, where he was a student in GomesTeixeira’s courses; then he joined the army school in Lisbon, reaching the rank of colonel. He was involved in thenational edition of the works of Pedro Nunes in Portugal and wrote on the history of mathematics in his country.See [26].

HMAT 26 RIEMANN’S IDEAS IN PORTUGAL 65

director of the Lisbon observatory and renowned for his knowledge of the history of theexact sciences in his country. His bibliography was published in 1900 as part of the Por-tuguese contribution to the Paris Exhibition. It was entitledLes mathematiques en Portu-gal au XIXeme siecle, aperc¸u historique et bibliographique[9]. Two other volumes werepresented by the Portuguese delegation; they gave a description and an account of the Por-tuguese colonial empire. In all three publications, one can detect the incipient influence ofnationalism on Portuguese intellectuals. This was a characteristic of the period, not unre-lated to the anti-French reaction. Later, Guimar˜aes published a more extended version ofhis work.

Guimaraes presented his work at the meeting of 10 August of the II Congr`es internationaldes math´ematiciens.17 The latter was one of several congresses sponsored by the Exhibition.It was indicated in the Congress’s report that Guimar˜aes gave references for 769 works in hismemoir; of these 226 were on analysis, 192 on geometry, and 351 on applied mathematics.He offered copies of his work to foreign participants. To this day, his detailed work has noparallel in other countries of the Iberian and Ibero-American world.

His attempt to encapsulate the history of mathematics in Portugal into a detailed andprecise compilation of dates, titles, and pages is a typical product of the school of historicalpositivism of the end of the 19th century which his local and French mentors followed. Italso shows the complexity of the breaking boundary with intellectual France. Positivism hadattracted a great deal of attention in Portugal.O Positivismo, founded in 1878, was amongthe leading philosophical journals of Portugal; it followed Laffitte’s orthodox version ofComte, as illustrated by Teixeira Bastos in his article “Do methodo positivo,” in the firstvolume ofO Positivismo.

7. FINAL REMARKS

Figueiredo did not continue publishing in mathematics or bringing more innovative ma-terial into print. Acceding fairly young to a chair in Coimbra, he may have found lecturingand personal influence more direct vehicles for bringing further novelties into his country.No doubt the university and the local scientific community of his time did not press himfor anything else. Further advances in his career would not be necessarily determined byhis scientific activities, unless these had a very exceptional international impact, as was thecase for Gomes Teixeira.

On a different level, this story also suggests that the process of transmission of mathe-matical ideas from leading to peripheral mathematical communities is a complex one, withselective sharp advances and retreats, and far from being a slow process of regular diffusion.It is also interesting to remark that Riemann’s ideas would not be transmitted successfully,at the level in which Figueiredo used them, from Portugal to neighboring Spain for severaldecades. Although Zoel Garcia de Galdeano wrote 30 pages or more on Riemann surfacesin hisTratado de analisis matematicoof 1904 (which were translated from Picard’sTraite

17 Ministere de commerce, de l’industrie, des postes et des t´elegraphes, Exposition universelle internationalede 1900. Direction g´enerale de l’exploitation, II Congr`es international des math´ematiciens, tenu `a Paris du 6 au12 Aout 1900, Proc`es-verbaux sommaires par M. E. Duporq, Ing´enieur des t´elegraphes, Secr´etaire general decongres, Paris, Imprimerie nationale, 1900, p. 16.

66 GRAY AND ORTIZ HMAT 26

d’analyse, vol. II, [18], and other sources), Julio Rey Pastor (1888–1962),18 some 30 yearsafter the publication of Figueiredo’s doctoral dissertation in Coimbra, could still refer tothis theory as one “which more urgently needs to be introduced in Spain” [21, 140].

ACKNOWLEDGMENTS

The authors thank the Director, Professor Manual Augusto Rodrigues, and the personnel of the Arquivo daUniversidade de Coimbra for their kind assistance in the researching of this paper. In particular, they thank Dr.Joao Manuel Saraiva de Carvalho, curator of the Arquivo; without his help this paper could not have been written.They also thank the referees for their comments, which have helped clarify several points in this paper. The secondauthor thanks the Royal Society, London, and The John Simon Guggenheim Memorial Foundation, New York forfinancial support while working on this paper.

REFERENCES

1. Joseph L. F. Bertrand,Traite de calcul differentiel et de calcul integral, Paris: Gauthier–Villars, 1870.

2. Charles A. A. Briot and Jean-Claude Bouquet,Theorie des fonctions doublement periodiques et, en particulier,des fonctions elliptiques, Paris: Mallet–Bachelier, 1859.

3. Charles A. A. Briot and Jean-Claude Bouquet,Theorie des fonctions elliptiques, Paris: Mallet–Bachelier,1875.

4. Alfred Clebsch and Carl L. F. von Lindemann,Vorlesungen ¨uber Geometrie, vol. 1, Leipzig: Teubner, 1876.

5. Heinrich Durege,Elemente der Theorie der Functionen einer complexen ver¨anderlichen Grosse mit beson-derer Berucksichtigung der Schopfungen Riemanns, Leipzig: Teubner, 1864.

6. Henrique Manuel de Figueiredo,Superficies de Riemann, Coimbra: Imprensa da Universidade, 1887.

7. Henrique Manuel de Figueiredo,Curvas planas algebricas, Coimbra: Imprensa da Universidade, 1888.

8. Zoel G. de Galdeano,Tratado de analisis matematico, Zaragoza: Emilio Casa˜nal, 1904.

9. Rudolpho Guimar˜aes,Les mathematiques en Portugal au XIXeme siecle, aperc¸u historique et bibliographique,Coimbra: Imprimerie de l’Universit´e, 1900; 2nd ed., 1909; microfiche ed., London: The Humboldt Library,1997.

10. Jakob L¨uroth, Beweis eines Satzes ¨uber rationale Curven,Mathematische Annalen9 (1876), 163–165.

11. Kenneth O. May,Bibliography and Research Manual of the History of Mathematics,Toronto: Univ. ofToronto Press, 1973.

12. Abilio Affonso da Silva Monteiro, Ora¸cao recitada pelo decano interino da Faculdade de Mathematica nodouturamento de Henrique Manuel de Figueiredo,O Instituto15 (1888), 260–265.

13. Carl G. Neumann,Vorlesungen ¨uber Riemann’s Theorie der Abel’schen Integrale, Leipzig, 1865; 2nd ed.,1884.

14. Eduardo L. Ortiz, ed.,The Works of Julio Rey Pastor, 8 vols., London: The Humboldt Society, 1988.

15. Eduardo L. Ortiz,El metodo de graffe en Espana, Logrono: University of La Rioja, 1987.

16. Eduardo L. Ortiz, Las relaciones cient´ıficas entre Argentina y Espa˜na, inII Encuentro hispano-argentino dehistoria de las ciencias, Madrid: Real Academia de Ciencias, 1990, pp. 339–356.

17. Eduardo L. Ortiz, The Nineteenth Century International Mathematical Community and Its Connection withThose on the Iberian Periphery, inL’Europe mathematique, Mathematical Europe, Catherine Goldstein,Jeremy Gray, Jim Ritter, ed., Paris:Editions de la maison des sciences de l’homme, 1996, pp. 321–343.

18. C.Emile Picard,Traite d’analyse, vol. 2, Paris: Gauthier–Villars, 1893.

19. Victor A. Puiseux, Recherches sur les fonctions alg´ebriques,Journal de mathematiques(1) 15 (1850),365–480.

20. Jos´e Maria de E¸ca de Queir´os,O Francesimo, Ultimas paginas, Porto, n.d.

18 On Rey Pastor, see [14] and the references given therein. He was the leading Spanish mathematician of histime, and learned the modern theory of complex variables in Germany, working under Carath´eodory and Koebe.

HMAT 26 RIEMANN’S IDEAS IN PORTUGAL 67

21. Julio Rey Pastor,Introduccion a la matematica superior, estado actual, metodos y problemas, Madrid 1916;reprinted in [14]; also reprinted, with a presentation by Eduardo L. Ortiz, Logro˜no: Instituto de estudiosriojanos, 1983.

22. Bernhard Riemann,Gesammelte mathematische Werke und wissenschaftliche Nachlass, ed. Heinrich Weber,Leipzig: Teubner, 1876.

23. Bernhard Riemann,Gesammelte Werke, Collected Papers, ed. R. Narasimhan, New York: Springer–Verlag,1990.

24. Manuel Augusto Rodrigues,Memoria professorum universitatis coimbrigensis, 1772–1938, vol. 2, Coimbra:Universidade de Coimbra, 1992.

25. George Salmon,Higher Plane Curves, 2nd ed., Dublin: Hodges, Foster & Co., 1879; German translationby Wilhelm Fiedler:Analytische Geometrie der hoheren ebenen Kurven, 2nd ed., Leipzig: Teubner, 1882;French translation by O. Chemin,Traite de geometrie analytique (courbes planes), et suivi d’uneetude surles points singuliers par G. Halphen, Paris, 1884.

26. Luis M. R. Saraiva, Historiography of Mathematics in the Works of Rodolfo Guimar˜aes,Historia Mathematica24 (1997), 86–97.

27. Francisco Gomes Teixeira,Traite des courbes speciales remarquables, Paris, 1892; reprint, New York:Chelsea, 1973.

28. Francisco Gomes Teixeira, reviews of [6; 7], inJornal des sciences mathematicas e astronomicas8 (1887–1888), 25, 183.